Three-body problem

In its traditional sense, the three-body problem is the problem of taking an initial set of data that specifies the positions, masses and velocities of three bodies for some particular point in time and then determining the motions of the three bodies, in accordance with the laws of classical mechanics (Newton's laws of motion and of universal gravitation).

In an extended modern sense, a three-body problem is a class of problems in classical or quantum mechanics that model the motion of three particles.

Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth and the Sun.[1]

The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his "Principia" (Philosophiæ Naturalis Principia Mathematica). In Proposition 66 of Book 1 of the "Principia", and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of the Earth and the Sun.

The problem became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea. This problem was addressed by Amerigo Vespucci and by Galileo Galilei before being solved by John Harrison's invention of the Marine chronometer. Before the chronometer became available, Vespucci had used, in 1499, knowledge of the position of the moon to determine his position in Brazil. However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun, and planets, on the motion of the Moon around the Earth.

Jean d'Alembert and Alexis Clairaut, who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality, and by the use of differential equations to be solved by successive approximations. They submitted their competing first analyses to the Académie Royale des Sciences in 1747.[2]

It was in connection with these researches, in Paris, in the 1740s, that the name "three-body problem" (Problème des Trois Corps) began to be commonly used. An account published in 1761 by Jean d'Alembert indicates that the name was first used in 1747.[3]

In 1887, mathematicians Ernst Bruns [4] and Henri Poincaré showed that there is no general analytical solution for the three-body problem given by algebraic expressions and integrals. The motion of three bodies is generally non-repeating, except in special cases.[5]

The circular restricted three-body problem is a valid approximation of elliptical orbits found in the Solar System, and this can be visualized as a combination of the potentials due to the gravity of the two primary bodies along with the centrifugal effect from their rotation (Coriolis effects are dynamic and not shown). The Lagrange points can then be seen as the five places where the gradient on the resultant surface is zero (shown as blue lines) indicating that the forces are in balance there.

A prominent example of the classical three-body problem is the movement of a planet with a satellite around a star. In most cases such a system can be factorized, considering the movement of the complex system (planet and satellite) around a star as a single particle; then, considering the movement of the satellite around the planet, neglecting the movement around the star. In this case, the problem is simplified to the two-body problem. However, the effect of the star on the movement of the satellite around the planet can be considered as a perturbation.

A three-body problem also arises from the situation of a spacecraft and two relevant celestial bodies, e.g. the Earth and the Moon, such as when considering a free return trajectory around the Moon, or other trans-lunar injection. While a spaceflight involving a gravity assist tends to be at least a four-body problem (spacecraft, Earth, Sun, Moon), once far away from the Earth when Earth's gravity becomes negligible, it is approximately a three-body problem.

Lagrange, tackling the general three-body problem, considered the behaviour of the distances between the bodies, without finding a general solution. But from his numerous equations he discovered two classes of constant-pattern solutions: collinear, in which one of the distances is the sum of the other two, and equiangular, in which the three distances are equal. Those classes yield what are now called L1, L2, L3, L4, and L5 respectively.

In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions, bringing the total number of families of repetitive motion to 16. One of the 16 families is a figure-eight pattern discovered in 1993 by physicist Cris Moore at the Santa Fe Institute.[5]

However this is possible only by taking certain assumptions which basically reduce the problem to a single-body problem within an energy potential. Generally even a two-body problem is not solvable analytically in quantum mechanics, since there is usually no analytical solution to the multi-particle Schrödinger partial differential equation. Some mathematical research in quantum mechanics is still dedicated either to finding a good numerical solution [8] or finding ways to reduce the problem into a more simple system that can be solved analytically such as the Hartree–Fock method and the Franck–Condon principle.

In 1912, the Finnish mathematician Karl Fritiof Sundman proved there exists a series solution in powers of t1/3 for the 3-body problem.[9] This series is convergent for all real t, except initial data that correspond to zero angular momentum. However, these initial data are not generic since they have Lebesgue measure zero.

An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore it is necessary to study the possible singularities of the 3-body problems. As it will be briefly discussed below, the only singularities in the 3-body problem are binary collisions (collisions between two particles at an instant), and triple collisions (collisions between three particles at an instant).

Now collisions, whether binary or triple (in fact any number), are somehow improbable—since it has been shown they correspond to a set of initial data of measure zero. However, there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So, Sundman's strategy consisted of the following steps:

Using an appropriate change of variables, to continue analyzing the solution beyond the binary collision, in a process known as regularization.

Prove that triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to L ≠ 0 he removed all real singularities from the transformed equations for the 3-body problem.

Showing that if L ≠ 0, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy's existence theorem for differential equations, there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (shades of Kovalevskaya).

Find a conformal transformation that maps this strip into the unit disc. For example if s = t1/3 (the new variable after the regularization) and if [clarification needed] then this map is given by:

This finishes the proof of Sundman's theorem.

Unfortunately the corresponding convergent series converges very slowly. That is, getting the value to any useful precision requires so many terms, that his solution is of little practical use. Indeed, in 1930 David Beloriszky calculated that if Sundman’s series were going to be used for astronomical observations then the computations would involve at least terms.[10]

The three-body problem is a special case of the n-body problem, which describes how n objects will move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Sundman for n = 3 and by Wang for n > 3 (see n-body problem for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes;[11] therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations (see n-body simulation). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum n-body problem. Among classical physical systems, the n-body problem usually refers to a galaxy or to a cluster of galaxies; planetary systems, such as star(s), planets, and their satellites, can also be treated as n-body systems. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.

^The 1747 memoirs of both parties can be read in the volume of Histoires (including Mémoires) of the Académie Royale des Sciences for 1745 (belatedly published in Paris in 1749) (in French):

Clairaut: "On the System of the World, according to the principles of Universal Gravitation" (at pp. 329–364); and

d'Alembert: "General method for determining the orbits and the movements of all the planets, taking into account their mutual actions" (at pp. 365–390).

The peculiar dating is explained by a note printed on page 390 of the 'Memoirs' section:"Even though the preceding memoirs, of Messrs. Clairaut and d'Alembert, were only read during the course of 1747, it was judged appropriate to publish them in the volume for this year" (i.e. the volume otherwise dedicated to the proceedings of 1745, but published in 1749).

^Jean d'Alembert, in a paper of 1761 reviewing the mathematical history of the problem, mentions that Euler had given a method for integrating a certain differential equation "in 1740 (seven years before there was question of the Problem of Three Bodies)": see d'Alembert, "Opuscules Mathématiques", vol.2, Paris 1761, Quatorzième Mémoire ("Réflexions sur le Problème des trois Corps, avec de Nouvelles Tables de la Lune ...") pp. 329–312, at sec. VI, p. 245.